1. **Problem Statement:** Simplify each algebraic expression by combining like terms. 2. **Important Rules:** - Like terms have the same variable raised to the same power. - Coefficients are the numerical parts of terms. - Constants are terms without variables. - To combine like terms, add or subtract their coefficients. 3. **Step-by-step Solutions:** **a.** $6x + 5x^2 + 2x + 11$ - Group like terms: $(6x + 2x) + 5x^2 + 11$ - Combine coefficients of $x$: $6x + 2x = \cancel{6x} + \cancel{2x} = 8x$ - Final simplified form: $$5x^2 + 8x + 11$$ **b.** $12g + 6g + 5g + 8$ - Group like terms: $(12g + 6g + 5g) + 8$ - Combine coefficients of $g$: $12g + 6g + 5g = \cancel{12g} + \cancel{6g} + \cancel{5g} = 23g$ - Final simplified form: $$23g + 8$$ **c.** $10f + 4 + 5f + 2e$ - Group like terms: $(10f + 5f) + 2e + 4$ - Combine coefficients of $f$: $10f + 5f = \cancel{10f} + \cancel{5f} = 15f$ - No like terms for $2e$ and $4$ - Final simplified form: $$15f + 2e + 4$$ **d.** $7r^2 + 2 + 3r^2 + 5$ - Group like terms: $(7r^2 + 3r^2) + (2 + 5)$ - Combine coefficients of $r^2$: $7r^2 + 3r^2 = \cancel{7r^2} + \cancel{3r^2} = 10r^2$ - Combine constants: $2 + 5 = 7$ - Final simplified form: $$10r^2 + 7$$ **e.** $2x + 4 + 6x + 3x^2 + 7$ - Group like terms: $(2x + 6x) + 3x^2 + (4 + 7)$ - Combine coefficients of $x$: $2x + 6x = \cancel{2x} + \cancel{6x} = 8x$ - Combine constants: $4 + 7 = 11$ - Final simplified form: $$3x^2 + 8x + 11$$